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1. Introduction

Sorting techniques have a wide variety of applications. Computer-Aided
Engineering systems often use sorting algorithms to help reason about geometric
objects, process numerical data, rearrange lists, etc. In general, therefore,
we will be interested in sorting a set of records containing keys,
so that the keys are ordered according to some well defined ordering rule,
such as numerical or alphabetical order. Often, the keys will form only
a small part of the record. In such cases, it will usually be more efficient
to sort a list of keys without physically rearranging the records.

2. Performance Criteria

There are several criteria to be used in evaluating a sorting algorithm:

Running time. Typically, an elementary sorting algorithm requires
O(N2)
steps to sort N randomly arranged items. More sophisticated
sorting algorithms require O(N log N) steps on average. Algorithms
differ in the constant that appears in front of the
N2
or N log N. Furthermore, some sorting algorithms are more
sensitive to the nature of the input than others. Quicksort, for
example, requires O(N log N) time in the average case, but requires
O(N2)
time in the worst case.

Memory requirements. The amount of extra memory required by a sorting
algorithm is also an important consideration. In place sorting
algorithms are the most memory efficient, since they require practically
no additional memory. Linked list representations require
an additional N words of memory for a list of pointers. Still other
algorithms require sufficent memory for another copy of the input array.
These are the most inefficient in terms of memory usage.

Stability. This is the ability of a sorting algorithm to preserve
the relative order of equal keys in a file.

Examples of elementary sorting algorithms are: selection sort, insertion
sort, shell sort and bubble sort. Examples of sophisticated sorting algorithms
are quicksort, radix sort, heapsort and mergesort. We will consider
a selection of these algorithms which have widespread use. In the
algorithms given below, we assume that the array to be sorted is stored
in the memory locations a[1],a[2],...,a[N]. The memory location
a[0]
is reserved for special keys called sentinels, which are described
below.

3. Selection Sort

This ``brute force'' method is one of the simplest sorting algorithms.

Approach:

Find the smallest element in the array and exchange it with the element
in the first position.

Find the second smallest element in the array and exchange it with the
element in the second position.

Selection sort is easy to implement; there is little that can go wrong
with it. However, the method requires O(N2) comparisons
and so it should only be used on small files. There is an important exception
to this rule. When sorting files with large records and small keys, the
cost of exchanging records controls the running time. In such cases, selection
sort requires O(N) time since the number of exchanges is at most
N.

4. Insertion Sort

This is another simple sorting algorithm, which is based on the principle
used by card players to sort their cards.

Approach:

Choose the second element in the array and place it in order with respect
to the first element.

Choose the third element in the array and place it in order with respect
to the first two elements.

Continue this process until done.

Insertion of an element among those previously considered consists of moving
larger elements one position to the right and then inserting the element
into the vacated position.

It is important to note that there is no test in the while loop to prevent
the index j from running out of bounds. This could happen if v
is smaller than a[1],a[2],...,a[i-1]. To remedy this situation,
we place a sentinel key in a[0], making it at least as small
as the smallest element in the array. The use of a sentinel is more
efficient than performing a test of the form while (j > 1 &&
a[j-1] > v). Insertion sort is an O(N2) method both
in the average case and in the worst case. For this reason, it is
most effectively used on files with roughly N < 20. However,
in the special case of an almost sorted file, insertion sort requires only
linear
time.

5. Shell Sort

This is a simple, but powerful, extension of insertion sort, which gains
speed by allowing exchanges of non-adjacent elements.

Definition:

An h-sorted file is one with the property that taking every
hth
element (starting anywhere) yields a sorted file.

Approach:

Choose an initial large step size, hK, and use insertion
sort to produce an hK-sorted file.

Choose a smaller step size, hK-1, and use insertion sort
to produce an hK-1-sorted file, using the hK-sorted
file as input.

Continue this process until done. The last stage uses insertion sort, with
a step size h1 = 1, to produce a sorted file.

Each stage in the sorting process brings the elements closer to their final
positions. The method derives its efficiency from the fact that insertion
sort is able to exploit the order present in a partially sorted input file;
input files with more order to them require a smaller number of exchanges.
It is important to choose a good sequence of increments. A commonly used
sequence is (3K-1)/2,...,121,40,13,4,1, which is obtained
from the recurrence hk = 3 hk+1+1. Note
that the sequence obtained by taking powers of 2 leads to bad performance
because elements in odd positions are not compared with elements in even
positions until the end.

Shell sort requires O(N3/2) operations in the worst
case, which means that it can be quite effectively used even for moderately
large files (say N < 5000).

6. Quicksort

This divide and conquer algorithm is, in the average case, the fastest
known sorting algorithm for large values of N. Quicksort is
a good general purpose method in that it can be used in a variety of situations.
However, some care is required in its implementation. Since the algorithm
is based on recursion, we assume that the array (or subarray) to be sorted
is stored in the memory locations a[left],a[left+1],...,a[right].
In order to sort the full array, we simply initialize the algorithm with
left
= 1 and right = N.

Approach:

Partition the subarray a[left],a[left+1],...,a[right] into two parts,
such that

element a[i] is in its final place in the array for some i
in the interval [left,right].

none of the elements in a[left],a[left+1],...,a[i-1] are greater
than a[i].

none of the elements in a[i+1],a[i+2],...,a[right] are less than
a[i].

Recursively partition the two subarrays, a[left],a[left+1],...,a[i-1]
and a[i+1],a[i+2],...,a[right], until the entire array is sorted.

How to partition the subarray a[left],a[left+1],...,a[right]:

Choose a[right] to be the element that will go into its final position.

Scan from the left end of the subarray until an element greater than a[right]
is found.

Scan from the right end of the subarray until an element less than a[right]
is found.

Exchange the two elements which stopped the scans.

Continue the scans in this way. Thus, all the elements to the left of the
left scan pointer will be less than a[right] and all the elements
to the right of the right scan pointer will be greater than a[right].

When the scan pointers cross we will have two new subarrays, one with elements
less than a[right] and the other with elements greater than a[right].
We may now put a[right] in its final place by exchanging it with
the leftmost element in the right subarray.

Note that this code requires a sentinel key in a[0] to stop the
right-to-left scan in case the partitioning element is the smallest element
in the file. Quicksort requires O(N log N) operations in the
average case. However, its worst case performance is O(N2),
which occurs in the case of an already sorted file! There are a number
of improvements which can be made to the basic quicksort algorithm.

Using the median of three partitioning method makes the worst case
far less probable, and it eliminates the need for sentinels. The basic
idea is as follows. Choose three elements, a[left], a[middle]
and a[right], from the left, middle and right of the array.
Sort them (by direct comparison) so that the median of the three is in
a[middle]
and the largest is in a[right]. Now exchange
a[middle]
with a[right-1]. Finally, we run the partitioning algorithm
on the subarray a[left+1],a[left+2],...,a[right-2] with
a[right-1]
as the partitioning element.

Another improvement is to remove recursion from the algorithm by
using an explicit stack. The basic idea is as follows. After
partitioning, push the larger subfile onto the stack. The smaller subfile
is processed immediately by simply resetting the parameters left
and right (this is known as end-recursion removal).
With the explicit stack implementation, the maximum stack size is about
log2
N. On the other hand, with the recursive implementation, the
underlying stack could be as large as N.

A third improvement is to use a cutoff to insertion sort whenever small
subarrays are encountered. This is because insertion sort, albeit
an O(N2) algorithm, has a sufficiently small constant
in front of the N2 to be more efficient than quicksort
for small N. A suitable value for the cutoff subarray size
would be approximately in the range 5 ~ 25.

7. Choosing a Sorting Algorithm

Table 1 summarizes the performance characteristics of some common sorting
algorithms. Shell sort is usually a good starting choice for moderately
large files N < 5000, since it is easily implemented. Bubble
sort, which is included in Table 1 for comparison purposes only, is generally
best avoided. Insertion sort requires linear time for
almost sorted
files, while selection sort requires linear time for files with large records
and small keys. Insertion sort and selection sort should otherwise
be limited to small files. Quicksort is the method to use for very
large sorting problems. However, its performance may be significantly
affected by subtle implementation errors. Furthermore, quicksort
performs badly if the file is already sorted. Another possible disadvantage
is that quicksort is not stable i.e. it does not preserve the relative
order of equal keys. All of the above sorting algorithms are in-place
methods. Quicksort requires a small amount of additional memory for
the auxiliary stack. There are a few other sorting methods which
we have not considered. Heapsort requires O(N log N) steps
both in the average case and the worst case, but it is about twice as slow
as quicksort on average. Mergesort is another O(N log N) algorithm
in the average and worst cases. Mergesort is the method of choice
for sorting linked lists, where sequential access is required.